Finite Group Databasebeta

← #20275855

Subgroups — #20275855

Order 6784.

290 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133216C211yesmaximal, characteristic
2C2×#13133142C211yesmaximal, characteristic
3C2×#13133140C211yesmaximal, characteristic
4#16810528C211yesmaximal, characteristic
5#16810385C211yesmaximal, characteristic
6#16810377C211yesmaximal, characteristic
7C53×#340C211yesmaximal, characteristic
8C22×Dic106—21no—
9C22×Dic106C2211yescharacteristic
10C22×#88322C2211yescharacteristic
11#13133216—22no—
12#13133216—22no—
13C2×C4⋊Dic53C2211yescharacteristic
14C2×#11719919—21no—
15C2×C4×Dic53C2211yescharacteristic
16#13133142—22no—
17#13133140—22no—
18#13133124—21no—
19#13133124—21no—
20C22×C8×C53C2211yescharacteristic
21C2×C4⋊C4×C53C2211yescharacteristic
22C2×C4⋊C4×C53C2211yescharacteristic
23C22×C4×C53—21no—
24C22×C4×C53—21no—
25C22×C4×C53C2311yescharacteristic
26C22×Dic53—21no—
27C22×Dic53—21no—
28C22×Dic53D411yescharacteristic
29C22×Dic53D411yescharacteristic
30C2×Dic106—21no—
31C2×Dic106—21no—
32C2×Dic106—21no—
33C2×Dic106—21no—
34C2×Dic106—24no—
35C2×Dic106—21no—
36C2×Dic106—42no—
37C2×Dic106—42no—
38C2×Dic106Q812yescomplemented
39C2×C8×C53—21no—
40C2×C8×C53—21no—
41C2×C8×C53—21no—
42C4⋊C4×C53—21no—
43C4⋊C4×C53—21no—
44C4⋊C4×C53—21no—
45C4⋊C4×C53D412yes—
46C4⋊Dic53—21no—
47C4⋊Dic53D412yes—
48#11719919—24no—
49C4×Dic53—21no—
50C4×Dic53—21no—
51C2×#88322—21no—
52C2×#88322—21no—
53C2×#88322—21no—
54C23×C53C2×D411yescharacteristic
55C2×C4×C53—22no—
56C2×C4×C53—41no—
57C2×C4×C53—41no—
58C2×C4×C53—41no—
59C2×C4×C53—41no—
60C2×C4×C53—41no—
61C2×C4×C53C2×D411yescharacteristic
62C2×C4×C53C2×D411yescharacteristic
63C2×C4×C53C2×Q811yescharacteristic
64C2×C4×C53#4111yescharacteristic
65C2×C4×C53#4111yescharacteristic
66C2×C4×C53#4111yescharacteristic
67C2×Dic53—21no—
68C2×Dic53—21no—
69C2×Dic53—21no—
70C2×Dic53—21no—
71C2×Dic53—22no—
72C2×Dic53—22no—
73C2×Dic53—41no—
74C2×Dic53—41no—
75C2×Dic53—41no—
76C2×Dic53—41no—
77C2×Dic53SD1612yes—
78C2×Dic53Q1612yes—
79Dic106—22no—
80Dic106—22no—
81Dic106—48no—
82Dic106—41no—
83Dic106—41no—
84C8×C53—41no—
85#88322—41no—
86C22×C53#12011yescharacteristic
87C22×C53C22⋊Q811yescharacteristic
88C22×C53#12411yescharacteristic
89C22×C53C2×SD1611yescharacteristic
90C22×C53C2×Q1611yescharacteristic
91C22×C53AGL(1,ℤ/8)11yescharacteristic
92C22×C53#13711yescharacteristic
93C4×C53—42no—
94C4×C53—81no—
95C4×C53C22⋊Q811yescharacteristic
96C4×C53C22⋊Q811yescharacteristic
97C4×C53#12411yescharacteristic
98C4×C53#12411yescharacteristic
99Dic53—22no—
100Dic53—22no—
101Dic53—42no—
102Dic53—42no—
103#2024—531nomaximal, Sylow
104C2×C53#39311yescharacteristic
105C2×C53#45011yescharacteristic
106C2×C53#45111yescharacteristic
107C2×C53#47511yescharacteristic
108C2×C53#47711yescharacteristic
109C2×C53#48711yescharacteristic
110C2×C53#48911yescharacteristic
111C2×C4⋊Q8—531no—
112C2×#103—531no—
113C2×#103—531no—
114#391—531no—
115#340—531no—
116#340D5311yescharacteristic
117#336—531no—
118C53#202411yescharacteristic, complemented, Sylow
119C22×Q8—531no—
120C22×Q8—1061no—
121C22×C8—531no—
122C22×C8C2×D5311yescharacteristic
123C4⋊Q8—1062no—
124C4⋊Q8—1062no—
125C2×C4⋊C4—531no—
126C2×C4⋊C4—1061no—
127C2×C4⋊C4C2×D5311yescharacteristic
128C2×C4⋊C4C2×D5311yescharacteristic
129C2×C42—531no—
130#103—1062no—
131#103—1062no—
132#95—1061no—
133#95—1061no—
134C2×Q8—532no—
135C2×Q8—1064no—
136C2×Q8—1061no—
137C2×Q8—1061no—
138C2×Q8—1061no—
139C2×Q8—1061no—
140C2×Q8—1061no—
141C2×Q8—2122no—
142C2×Q8—2122no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—531no—
146C22×C4—531no—
147C22×C4—1061no—
148C22×C4—1061no—
149C22×C4C22×D5311yescharacteristic
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—1061no—
154C2×C8—1061no—
155C2×C8—1061no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—532no—
160C4⋊C4—1064no—
161C4⋊C4—1061no—
162C4⋊C4D21212yes—
163C42—1061no—
164C42—1061no—
165C23D4×D5311yescharacteristic
166Q8—1062no—
167Q8—1062no—
168Q8—2128no—
169Q8—2121no—
170Q8—2121no—
171C2×C4—22no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—532no—
178C2×C4—532no—
179C2×C4—1061no—
180C2×C4—1061no—
181C2×C4—1061no—
182C2×C4—1061no—
183C2×C4—1062no—
184C2×C4—1062no—
185C2×C4—2121no—
186C2×C4—2121no—
187C2×C4—2121no—
188C2×C4—2121no—
189C2×C4C2×D21211yescharacteristic
190C2×C4#1171994511yescharacteristic
191C2×C4#1171994511yescharacteristic
192C2×C4D4×D5311yescharacteristic
193C2×C4#1171994711yescharacteristic
194C2×C4Q8×D5311yescharacteristic
195C8—41no—
196C8—2121no—
197C22#1313318711yescharacteristic
198C22#1313319511yescharacteristic
199C22#1313320311yescharacteristic
200C22SD16×D5311yescharacteristic
201C22Q16×D5311yescharacteristic
202C22#1313324111yescharacteristic
203C22#1313324611yescharacteristic
204C4—42no—
205C4—81no—
206C4—1062no—
207C4—1062no—
208C4—2122no—
209C4—2122no—
210C4#1313319511yescharacteristic
211C4#1313319711yescharacteristic
212C4#1313320311yescharacteristic
213C4#1313320511yescharacteristic
214C2#1681053611yescharacteristic
215C2#1681077511yescharacteristic
216C2#1681078011yescharacteristic
217C2#1681090711yescharacteristic
218C2#1681091711yescharacteristic
219C2#1681097211yescharacteristic
220C2#1681098111yescharacteristic
subgroupG/H
centerC23D4×D53
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#340C2
socleC23×C53C2×D4
hypercenter#340D53
hypocenterC53#2024
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
38. C2×Dic106Q8semidirect
118. C53#2024semidirect